[csabatka]

I know that the cables connecting the screen should not be bent back farther than 90 degrees while replacing the battery of an iPhone 5s. I've seen videos of people keeping the screen connected while replacing the battery of an iPhone 5, but in the videos of people replacing the iPhone 5s battery they disconnect and remove the screen. Is removal of the screen necessary to replace an iPhone 5s battery?

[Tartarus]

It would be ideal if you just went to the Apple Store, paid them $79 and have them replace it for you. (When I did that, they simply replaced my whole device for that same money).

If you want to replace it yourself, it's best to remove the screen from the housing, to avoid any tear and damage to the connector cable.

ifixit has instruction videos for this.